Hersh correctly notes the essential limit of logicism -- the Axiom of Infinity
is not generally
accepted as a principle of logic. But of course a corollary is that logicism
is a partial
success -- all the mathematics which can be formalized without this axiom is
indeed philosophically certain (and this is a lot of mathematics!).
It's remarkable that practically all the rest of mathematics follows from just
two additional axioms (Infinity and Choice). Furthermore, it is possible to
make a case for these axioms on philosophical and logical grounds (though not
a universally convincing one). I think it
would be fruitful to discuss whether these two principles should properly be
regarded as
"mathematical" or "logical".